External or internal shocks may lead to the collapse of a system consisting
of many agents. If the shock hits only one agent initially and causes it to
fail, this can induce a cascade of failures among neighoring agents. Several
critical constellations determine whether this cascade remains finite or
reaches the size of the system, i.e. leads to systemic risk. We investigate the
critical parameters for such cascades in a simple model, where agents are
characterized by an individual threshold \theta_i determining their capacity to
handle a load \alpha\theta_i with 1-\alpha being their safety margin. If agents
fail, they redistribute their load equally to K neighboring agents in a regular
network. For three different threshold distributions P(\theta), we derive
analytical results for the size of the cascade, X(t), which is regarded as a
measure of systemic risk, and the time when it stops. We focus on two different
regimes, (i) EEE, an external extreme event where the size of the shock is of
the order of the total capacity of the network, and (ii) RIE, a random internal
event where the size of the shock is of the order of the capacity of an agent.
We find that even for large extreme events that exceed the capacity of the
network finite cascades are still possible, if a power-law threshold
distribution is assumed. On the other hand, even small random fluctuations may
lead to full cascades if critical conditions are met. Most importantly, we
demonstrate that the size of the "big" shock is not the problem, as the
systemic risk only varies slightly for changes of 10 to 50 percent of the
external shock. Systemic risk depends much more on ingredients such as the
network topology, the safety margin and the threshold distribution, which gives
hints on how to reduce systemic risk.Comment: 23 pages, 7 Figure
